Locally maximally mixed states
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Locally maximally mixed states Lin Chen1,2 · Mengyao Hu1 Received: 22 February 2020 / Accepted: 7 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Preparing the locally maximally mixed (LMM) states is a physically operational work. We investigate the set Pd containing two-qudit LMM states. We show that the point with a canonical decomposition (CD) has either the unique or infinitely many CDs. Next we show that the point in P2 has infinitely many CDs. Further we construct the necessary and sufficient condition by which the non-extreme point of rank two has the unique CD. We also show that the maximally correlated state of rank d is not an extreme point of Pd . As an application, we show that if the range of rank-three ρ ∈ P3 is spanned by product vectors, then ρ is not an extreme point of P3 . Moreover, ρ is realizable by unitary channels as a method of constructing a family of two-qutrit LMM states. We also prove that Conjecture 1 in [C. King et al., J. Phys. A: Math. Theor 40, 7939 (2007)] holds for ρ.
1 Introduction In quantum physics, bipartite mixed states are the convex sum of pure states. Mixed states exist more generally than pure states due to the unavoidable noise from nature, which may be characterized by completely positive maps in quantum physics. Locally maximally mixed (LMM) states, physically defined as the mixed states with reduced density operators being the maximally mixed states, have been extensively investigated in the past decades. They include the pure maximally entangled states (MESs) such as the two-qubit Bell states, and mixed states such as Werner states, isotropic states, Bell diagonal states, etc [1–5]. They play a key role in many applications such as quantum teleportation [6] and cryptography [7–10], as well as fundamental research
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Mengyao Hu [email protected] Lin Chen [email protected]
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School of Mathematical Sciences, Beihang University, Beijing 100191, China
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International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China 0123456789().: V,-vol
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L. Chen, M. Hu
like entanglement distillation [11,12]. As a result, the preparation and understanding of such mixed states is a basic task for quantum information processing. In particular, the MES has been widely investigated by theory and experiment [13–16]. Interestingly, it has been shown that the LMM state remains entangled when the dimension of pure states is larger than or equal to the number of entangled pure states to be mixed [17]. In the quantum internet [18], it is of pivotal importance to establish entanglement between legal parties [19] if the legal users of the network having different priority levels is necessary. Recently, a class of Bell inequalities that are tailored has been introduced to detect MESs for device-independent protocols [20]. It is known that MESs are interconvertible under local unitary operations. The latter form a small subset of local operations and classical communications (LOCC)
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